2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156845This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-folds. The result is similar to the fact that birational Calabi-Yau 3-folds have the same betti numbers or Hodge numbers.Some explanations and proofs are added. To appear in Communications in Number Theory and PhysicsAlgebraic Geometry14E30, 14D20, 18E30,text